Subgroup ($H$) information
| Description: | $D_{70}:C_{10}$ |
| Order: | \(1400\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rr}
90 & 0 \\
0 & 153
\end{array}\right), \left(\begin{array}{rr}
79 & 0 \\
0 & 249
\end{array}\right), \left(\begin{array}{rr}
170 & 0 \\
0 & 81
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 86
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 128
\end{array}\right), \left(\begin{array}{rr}
0 & 1 \\
1 & 0
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $D_{280}:C_{10}$ |
| Order: | \(5600\)\(\medspace = 2^{5} \cdot 5^{2} \cdot 7 \) |
| Exponent: | \(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{140}.C_6.C_2^3.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_4\times C_7:(C_2\times C_6\times F_5)$ |
| $W$ | $D_{70}$, of order \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_2\times D_{140}$ |